Notice that for a fixed <math>\alpha</math>, the ''t-critical values'' (for any degree-of-freedom) exceeds the [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Mean:_Large_Samples |corresponding normal z-critical values]], which are used int he large-sample interval estimation.

Notice that for a fixed <math>\alpha</math>, the ''t-critical values'' (for any degree-of-freedom) exceeds the [[AP_Statistics_Curriculum_2007#Estimating_a_Population_Mean:_Large_Samples |corresponding normal z-critical values]], which are used int he large-sample interval estimation.

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* For <math>\alpha=0.1</math>, the <math>80% CI(\mu)</math> is constructed by:

* For <math>\alpha=0.1</math>, the <math>80% CI(\mu)</math> is constructed by:

Now we discuss estimation when the sample-sizes are small, typically < 30 observations. Naturally, the point estimates are less precise and the interval estimates produce wider intervals, compared to the case of large-samples.

Point Estimation of a Population Mean

The population mean may also be estimated by the sample average using a small sample. That is the sample average , constructed from a random sample of the procees {}, is an unbiased estimate of the population mean μ, if it exists! Note that the sample average may be susceptible to outliers.

Interval Estimation of a Population Mean

For small samples, interval estimation of the population means (or Confidence intervals) are constructed as follows. Choose a confidence level (1 − α)100%, where α is small (e.g., 0.1, 0.05, 0.025, 0.01, 0.001, etc.). Then a (1 − α)100% confidence interval for μ is defined in terms of the T-distribution:

The margin of error E is defined as

The Standard Error of the estimate is obtained by replacing the unknown population standard deviation by the sample standard deviation:

is the critical value for the T(df=sample-size -1) distribution distribution at .

Example

To parallel the example in the large sample case, we consider again the number of sentences per advertisement as a measure of readability for magazine advertisements. A random sample of the number of sentences found in 10 magazine advertisements is listed. Use this sample to find point estimate for the population mean μ.

16

9

14

11

17

12

99

18

13

12

A confidence interval estimate of μ is a range of values used to estimate a population parameter (interval estimates are normally used more than point estimates because it is very unlikely that the sample mean would match exactly with the population mean) The interval estimate uses a margin of error about the point estimate.

Before you find an interval estimate, you should first determine how confident you want to be that your interval estimate contains the population mean.

Hands-on activities

Sample statistics, like the sample-mean and the sample-variance, may be easily obtained using SOCR Charts. The images below illustrate this functionality (based on the Bar-Chart and Index-Chart) using the 30 observations of the number of sentences per advertisement, reported above.